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Chapter 9 Life and Times on the Main Sequence In the preceding chapter we considered the collapse of a protostar to the main sequence. In this chapter we examine the nature of life on the main sequence for such a star. • The essence of life on the main sequence is stable burning of H into He in hydrostatic equilibrium, by – PP chains for stars of a solar mass or less, and – the CNO cycle for more massive main-sequence stars. • Since we have examined in some detail hydrostatic equilibrium, energy production by the PP chains and the CNO cycle, we already understand the essence of life on the main sequence. It is appropriate that we begin by examining this main sequence star that we know the best: the Sun. 327 328 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 9.1 The Standard Solar Model The Sun is by far the most studied star. We have a large amount of relevant data, and considerable understanding of how the Sun functions. This has allowed the construction of a Standard Solar Model: Standard Solar Model: a mathematical model of the Sun that uses 1. fundamental knowledge from fields such as nuclear and atomic physics, 2. measured key quantities, and 3. a few assumptions to describe all solar observations. Standard Solar Models are important because • They fix the Sun’s helium abundance and the convection length scale in the solar surface. • They provide a benchmark for measuring improved solar modeling and a starting point for more general stellar modeling. 9.1. THE STANDARD SOLAR MODEL The essence of the Standard Solar Model is that a 1 M⊙ ZAMS star is evolved to the present age of the Sun subject to the following assumptions: 1. The Sun was formed from a homogeneous mixture of gases. 2. The Sun is powered by nuclear reactions in its core. 3. The Sun is approximately in hydrostatic equilibrium, with the gravitational forces that attempt to compress it almost exactly compensated by forces arising from gradients in internal gas and radiation pressure. 4. Some deviation from equilibrium is permitted as the Sun evolves, but these are assumed to be small and slow. 5. Energy is transported from the core, where it is produced, to the surface, where it is radiated into space • by photons (radiative transport) • by large-scale vertical motion of packets of hot gas (convection). 6. The role of conduction in heat transport is considered negligible. Let us now discuss each of these assumptions that enter the Standard Solar Model in a little more depth. 329 330 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 9.1.1 Composition of the Sun • The assumption that the Sun was formed from a homogenous mixture of gases is motivated by the strong convection expected in the protostar during collapse to the main sequence. • The surface abundances are then assumed to have been undisturbed in the subsequent evolution, so that present surface abundances indicate the composition of the original solar core. • The abundance of most elements in the surface can be inferred by spectroscopy. The exception is the noble gases He, Ne, and Ar. They are not excited significantly by the blackbody emission spectrum of the photosphere, so their abundance cannot be determined well from the spectrum. • Because evolution of the Sun’s luminosity depends on the mean molecular weight raised to the power 7.5 (see Exercise 9.1), which is strongly influenced by the helium abundance, the H/He ratio is normally taken as an adjustable parameter in solar models. • The H/He ratio is determined by requiring that the luminosity of the Sun at the present age of the Solar System (4.6 billion years, as determined by dating meteorites) be accurately reproduced by the model. 9.1. THE STANDARD SOLAR MODEL 9.1.2 Nuclear Energy Generation and Composition Changes The Sun is assumed to derive its power and associated composition changes from the proton–proton chains PP-1, PP-2, and PP-III, and the CNO cycle. The nuclear reaction networks describing this energy and element production are solved by • dividing the Sun into concentric shells, • calculating the nuclear reactions in each shell as a function of the current temperature and density there, and • using the updated composition and the energy production as constant input to the partial differential equations describing the solar equilibrium structure. 331 332 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 9.1.3 Hydrostatic Equilibrium Since the dynamical timescale of the Sun is less than an hour, τhydro ≃ (Gρ̄ )1/2 ≃ 55 minutes, the Sun may be expected to have reached hydrostatic equilibrium quickly. • However, a Standard Solar Model allows small expansions and contractions in response to time evolution of the star. • Re-equilibration is assumed to be very fast compared with the timescale for evolution. • The pressure is – composed of both gas pressure and radiation pressure, but – the radiation pressure even at the center is only about 0.05% of the total pressure. • A Standard Solar Model typically ignores the effects of rotation and magnetic-field pressure on hydrostatic equilibrium. • Likewise, stellar pulsations are neglected in determining hydrostatic equilibrium. 9.1. THE STANDARD SOLAR MODEL 9.1.4 Energy Transport It is assumed that (1) energy transport in the Sun by acoustic or gravitational waves is negligible, and that (2) the energy produced internally in the Sun is transported by radiative diffusion and convection to the surface. • In the interior, transport is assumed to be by radiative diffusion unless the critical gradient for convective instability is exceeded, in which case the Sun transports energy in that region convectively with an adiabatic temperature gradient. • In the surface region, the actual gradient is steeper than the adiabatic gradient and mixing length theory is used to model convection. • Because convection in the surface region is difficult to calculate reliably, the mixing length in units of the scale height is taken as an adjustable parameter, to be fixed by requiring the model to yield the observed radius of the Sun. • The opacities required for radiative diffusion of energy are Rosseland mean opacities that must be calculated numerically. They are among the least well-determined quantities entering the Standard Solar Model, with typical uncertainties in the 10–20% range. 333 334 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE Optical depth and the solar surface The optical depth τ at radius r is defined in terms of the radiative opacity κ by τ= Z ∞ r κρ dr. It measures the probability that photons interact with solar material before being radiated into space. • The radius of the Sun is defined to be that distance from the center where the optical depth is 23 . • The diffusion approximation for radiative transport fails when τ is lower than about 1–10 because the mean free path for photons then becomes very long • (In the solar surface, the mean free path for photons is of order 107 cm or longer, compared with fractions of a cm in the interior). • The region of the solar surface where optical depth is less than about 1 is called the solar atmosphere. • Methods used to deal with radiative transport in the solar atmosphere are much more complicated that those adequate for the solar interior because one can no longer make a diffusion approximation. • It is essential to model the atmosphere adequately because it defines outer boundary conditions and this is where the solar spectrum is produced. 9.1. THE STANDARD SOLAR MODEL 335 9.1.5 Constraints and Solution Solution of the Standard Solar Model problem corresponds to • evolving in time four partial differential equations in five unknowns (P, T , r, m(r), and L), • supplemented by an equation of state and equations governing composition change (one for each species), • subject to constraints on calculated radius, luminosity, and mass. Two equations governing hydrostatic equilibrium, dm = 4π r2 ρ (r) dr Gm(r) dP =− 2 ρ dr r Mass conservation Hydrostatic equilibrium, three equations for luminosity and temperature gradients, dL = 4π r2ε (r) dr dT 3ρ (r)κ (r) L(r) =− dr 4acT 3 (r) 4π r2 dT γ − 1 T dP = dr γ P dr Luminosity (If radiative) (If convective), equations governing composition changes, dn 1 =− n dt τ Nucleosynthesis, and an equation of state, P = P(T, ρ , Xi , . . .) Equation of state. 336 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE The network of equations required to describe nuclear energy and element production is solved separately for each timestep in each zone. • The equation of state is assumed to be given by the ideal gas law for regions that are completely ionized. • Otherwise, a numerical equation of state calculated at Lawrence Livermore National Laboratory is typically used. • The Standard Solar Model solution is constructed iteratively. • Starting values for the helium abundance and the mixing length parameter are used to evolve an initial zero-age model to the current age of the Sun. • The model’s luminosity and radius are then compared with observations, the helium abundance and mixing length parameters adjusted accordingly, and the model is evolved again. • This cycle is repeated until convergence is obtained. Table 9.1 gives the temperature, density, pressure, and luminosity of a Standard Solar Model as a function of radius and enclosed mass at that radius. 9.1. THE STANDARD SOLAR MODEL 337 Table 9.1: A Standard Solar Model M/M⊙ R/R⊙ T (K) ρ (g cm−3 ) P (dyn cm−2 ) L/L⊙ 0.0000298 0.00650 1.568E+07 1.524E+02 2.336E+17 0.00027 0.0008590 0.02005 1.556E+07 1.483E+02 2.280E+17 0.00753 0.0065163 0.04010 1.516E+07 1.359E+02 2.111E+17 0.05389 0.0207399 0.06061 1.456E+07 1.193E+02 1.868E+17 0.15638 0.0439908 0.08041 1.386E+07 1.027E+02 1.606E+17 0.29634 0.0762478 0.10006 1.310E+07 8.729E+01 1.349E+17 0.45135 0.1173929 0.12000 1.231E+07 7.350E+01 1.108E+17 0.60142 0.1672004 0.14056 1.150E+07 6.123E+01 8.892E+16 0.73152 0.2203236 0.16027 1.076E+07 5.114E+01 7.094E+16 0.82657 0.2800107 0.18104 1.002E+07 4.205E+01 5.517E+16 0.89658 0.3393826 0.20107 9.353E+06 3.459E+01 4.279E+16 0.94011 0.3966733 0.22038 8.762E+06 2.847E+01 3.319E+16 0.96616 0.4559683 0.24084 8.188E+06 2.301E+01 2.516E+16 0.98259 0.5114049 0.26085 7.676E+06 1.857E+01 1.907E+16 0.99183 0.5627338 0.28058 7.214E+06 1.496E+01 1.446E+16 0.99669 0.6099028 0.30016 6.794E+06 1.203E+01 1.096E+16 0.99860 0.6564038 0.32132 6.379E+06 9.484E+00 8.119E+15 0.99941 0.6952616 0.34091 6.028E+06 7.605E+00 6.156E+15 0.99976 0.7304369 0.36063 5.703E+06 6.092E+00 4.667E+15 0.99993 0.7621708 0.38053 5.400E+06 4.876E+00 3.539E+15 1.00002 0.7907148 0.40067 5.117E+06 3.900E+00 2.683E+15 1.00005 0.8163208 0.42109 4.851E+06 3.118E+00 2.034E+15 1.00007 0.8374222 0.44008 4.621E+06 2.539E+00 1.578E+15 1.00007 0.8580756 0.46112 4.383E+06 2.029E+00 1.197E+15 1.00006 0.8750244 0.48072 4.176E+06 1.651E+00 9.287E+14 1.00006 0.8902432 0.50063 3.978E+06 1.345E+00 7.206E+14 1.00005 0.9038831 0.52086 3.789E+06 1.095E+00 5.591E+14 1.00004 0.9160850 0.54139 3.606E+06 8.924E-01 4.339E+14 1.00004 0.9260393 0.56033 3.445E+06 7.413E-01 3.445E+14 1.00003 0.9358483 0.58142 3.273E+06 6.052E-01 2.673E+14 1.00003 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 338 Table 9.1: (Continued) Standard Solar Model M/M⊙ R/R⊙ T (K) ρ (g cm−3 ) P (dyn cm−2 ) L/L⊙ 0.9438189 0.60081 3.120E+06 5.040E-01 2.123E+14 1.00002 0.9509668 0.62036 2.969E+06 4.205E-01 1.686E+14 1.00002 0.9573622 0.64001 2.818E+06 3.517E-01 1.339E+14 1.00002 0.9636045 0.66168 2.648E+06 2.900E-01 1.039E+14 1.00001 0.9686223 0.68129 2.485E+06 2.445E-01 8.249E+13 1.00001 0.9730081 0.70042 2.315E+06 2.081E-01 6.572E+13 1.00001 0.9771199 0.72033 2.115E+06 1.780E-01 5.161E+13 1.00001 0.9811002 0.74162 1.899E+06 1.513E-01 3.936E+13 1.00000 0.9842836 0.76050 1.718E+06 1.299E-01 3.055E+13 1.00000 0.9874435 0.78148 1.526E+06 1.085E-01 2.264E+13 1.00000 0.9900343 0.80103 1.355E+06 9.066E-02 1.678E+13 1.00000 0.9922832 0.82051 1.193E+06 7.470E-02 1.215E+13 1.00000 0.9942853 0.84082 1.031E+06 5.987E-02 8.406E+12 1.00000 0.9958822 0.86022 8.826E+05 4.733E-02 5.682E+12 1.00000 0.9972278 0.88035 7.356E+05 3.590E-02 3.585E+12 1.00000 0.9982619 0.90020 5.966E+05 2.613E-02 2.110E+12 1.00000 0.9990296 0.92017 4.627E+05 1.775E-02 1.107E+12 1.00000 0.9995498 0.94015 3.343E+05 1.080E-02 4.833E+11 1.00000 M⊙ = 1.989 × 1033 g R⊙ = 6.96 × 1010 cm L⊙ = 3.827 × 1033 erg s−1 9.1. THE STANDARD SOLAR MODEL Figure 9.1 illustrates graphically some of the parameters of this model plotted versus the radius and Fig. 9.2 plots the same quantities versus the enclosed mass coordinate. 339 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 340 1.2 108 1.0 107 T(K) M/M 0.8 0.6 106 0.4 0.2 0 0 0.2 0.4 0.6 0.8 105 1.0 0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 0.6 0.8 1.0 R/R R/R 1018 160 1017 1016 P (dyne/cm2) ρ (g/cm3) 120 80 1015 1014 1013 40 1012 0 0 0.2 0.4 0.6 0.8 1011 1.0 0 0.2 0.4 R/R R/R 1.2 1.0 Luminosity 1.0 0.8 Mass fraction L/L 0.8 0.6 Derivative/10 0.4 0.6 0.4 4He 0.2 0.2 0 1H 3He (x 100) 0 0.2 0.4 0.6 R/R 0.8 1.0 0 0 0.2 0.4 R/R Figure 9.1: Parameters from a Standard Solar Model (Table 9.1) plotted versus the radial coordinate. 9.1. THE STANDARD SOLAR MODEL 341 1.2 108 1.0 107 T(K) R/R 0.8 0.6 106 0.4 0.2 0 0 0.2 0.4 0.6 0.8 105 1.0 0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 M/M M/M 1018 160 1017 1016 P (dyne/cm2) ρ (g/cm3) 120 80 1015 1014 1013 40 1012 0 0 0.2 0.4 0.6 0.8 1011 1.0 0 0.2 0.4 M/M M/M 1.2 1.0 1.0 0.8 Mass fraction L/L 0.8 0.6 0.4 0.6 0.4 4He 0.2 0.2 0 1H (x 100) 3He 0 0.2 0.4 0.6 M/M 0.8 1.0 0 0 0.2 0.4 0.6 0.8 1.0 M/M Figure 9.2: Parameters from a Standard Solar Model (Table 9.1) plotted versus the enclosed mass. 342 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE The Standard Solar Model may be tested by comparing its predictions with observations. • These tests range from general ones, such as accounting for the existence, age, and energy output of the Sun, to specific ones such as the accounting for the results of solar seismology. • The Standard Solar Model has passed these tests very well. We now discuss two examples of how the Standard Solar Model description of the solar interior can be tested: • The subsurface structure as inferred from helioseismology and • The spectrum and overall flux of neutrinos emitted from the solar core. 9.2. HELIOSEISMOLOGY 9.2 Helioseismology One way to study the Sun’s interior is to study the propagation of waves in its body. • This is similar to the way geologists learn about the interior of the Earth by studying seismic waves or how we may infer the composition of a bell by striking it and studying the sound frequencies that it produces. • The corresponding field of study is called helioseismology. 343 344 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 9.2.1 p-Modes Solar oscillations were discovered by studying Doppler shifts of surface absorption lines. • It was found that the solar surface consists of patches oscillating on a timescale of five minutes with a velocity amplitude of 0.5 km s−1. • These 5-minute oscillations represent pressure waves (p-modes) trapped between the surface and the lower boundary of the convective zone. • They are reflected from the solar surface by density gradients. • They are refracted near the bottom of the convection zone because of changing soundspeed in that region. 9.2. HELIOSEISMOLOGY 9.2.2 g-Modes In addition to p-modes associated with acoustical waves trapped near the solar surface, the Sun may also exhibit g-modes: • These correspond to oscillations in which the restoring force is gravity. • If g-modes can be observed, they carry information about much deeper regions of the Sun than that carried by the p-modes. 345 346 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE km/s -2 -1 0 1 2 Figure 9.3: Doppler velocity field for the Sun. 9.2.3 Surface Vibrations and the Solar Interior One can learn about waves in the interior by dividing the surface of the Sun up into small regions and determining the radial velocity of each region from Doppler shifts. • The result is called the velocity field for the Sun (Fig. 9.3). • A diagram showing how the velocity field varies across the solar disk is called a dopplergram. • Superposed on the overall rotation are fluctuations corresponding to vertical motion of the Sun’s surface in localized regions. • The Michelson Doppler Imager (MDI) instrument on the SOHO observatory orbits the Sun 1.5 million km sunward from Earth. • It is capable of measuring vertical displacement on the solar surface at a million points simultaneously and can detect displacement velocities as small as 1 mm s−1. 9.2. HELIOSEISMOLOGY The Sun vibrates at a complex set of frequencies, with the dominant frequency corresponding to the 5-minute oscillation described above. • By decomposing the observed vibration of the Sun into a superposition of standing acoustic waves, it is possible to learn about the interior. • Such decompositions indicate that the observed motion of the surface is a superposition of several million resonant modes with different frequencies and horizontal wavelengths. • Individual modes in this decomposition may have velocity amplitudes as large as 20 cm s−1 and 1–2 meter vertical displacements. Presently, helioseismology is placing strong constraints on our theories of the solar interior. • The analysis is complex but the basic idea is simple: changes in the properties of the solar interior (for example, the amount of helium in some region) affects the way sound waves travel through the interior. • This will in turn influence the way the surface vibrates. 347 348 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE Two important pieces of information obtained from early helioseismology are that • the abundance of helium in the interior (but outside the core) is about the same as at the surface, and that • convection extends about 30 percent of the way to the center. We discuss in the next two sections information from helioseismology concerning sound speed in the solar interior and the rotation of the solar interior. 9.2. HELIOSEISMOLOGY 0.000 Shear ? ∆ s 2/ s 2 0.002 Convection Energy Production 0.004 349 -0.002 Radiative -0.004 0 0.2 0.4 0.6 0.8 1 R/R Figure 9.4: Deviation of the sound speed inferred using helioseismology from that predicted by the Standard Solar Model. 9.2.4 Speed of Sound in the Sun The speed of sound is sensitive to internal properties. • Figure 9.4 illustrates information about the speed of sound inside the Sun as inferred from SOHO helioseismology. • The data are shown in the form of the fractional deviation of the square of the sound speed s from that expected from the Standard Solar Model, with zero indicating perfect agreement. • The finite extent of the shaded area indicates the uncertainty in the measurement. CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 0.002 Convection Energy Production 0.004 0.000 Shear ? ∆ s 2/ s 2 350 -0.002 Radiative -0.004 0 0.2 0.4 0.6 0.8 1 R/R • The deviations are very small (fraction of a percent). • Largest deviations are concentrated in two regions: – a region just below the convection boundary at about 60–70 percent of the solar radius and – a region in the energy-producing zone near about 20 percent of the solar radius. • Both deviations are thought to result from small differences in helium concentration from that predicted by the accepted models of the Sun. • Negative deviations imply higher helium concentration; positive imply reduced helium concentration. 9.2. HELIOSEISMOLOGY 351 Rotation Rate (nHz) Convection Zone 0 Surface Doppler 500 o 0 450 30 400 60 o o 30 o o 60 o 350 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R/R Figure 9.5: Rotation rates at different latitudes in the Sun as a function of radius, as inferred from helioseismology with SOHO. Near the surface the rotation rate is different at different latitudes but inside the convection zone the rotation rates at all latitudes become similar. 9.2.5 Internal Rotation Rate of the Sun Data from SOHO helioseismology have also been used to deduce the internal rotation rate of the Sun. Figure 9.5 illustrates rotation rates inferred from solar vibrational data. • Results are plotted for three different latitudes versus the fraction of the solar radius; widths of the curves indicate the estimated uncertainties in the measurements. • The surface velocities at these latitudes as measured directly by Doppler shifts are indicated by the arrows on the right side. • We notice the differential rotation of the Sun near the surface (different latitudes rotating at different rates), as was already known from surface observations. CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 352 Rotation Rate (nHz) Convection Zone 0 Surface Doppler 500 o 0 450 30 400 60 o o 30 o o 60 o 350 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R/R • But the helioseismology data indicate that inside the convective zone, beginning at about 65 percent of the solar radius, the Sun begins to rotate almost as a rigid body. • That is, the curves for the three different latitudes converge to similar values, indicating that all latitudes are rotating with about the same rate. 9.3. SOLAR NEUTRINO PRODUCTION 9.3 Solar Neutrino Production Helioseismology allows us to probe the interior of the Sun. A second way in which we can study the (deep) interior of the Sun is by detecting the neutrinos that are produced there. • The energy powering the surface photon luminosity must make its way on a 100,000-year or greater timescale to the surface before being radiated. • Neutrinos emitted from the core are largely unimpeded in their exit from the Sun, reaching the Earth 8.5 minutes after they were produced. Therefore, neutrinos carry immediate and more direct information about the current conditions in the solar core than do the photons emitted from the solar photosphere. 353 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 354 • There are eight reactions or decays that are of some significance in solar energy production that may produce neutrinos: pp p + p → 2H + e+ + νe Q ≤ 0.420 MeV pep p + e− + p → 2H + νe Q = 1.442 MeV hep 3 He + p → 4 He + ν Q ≤ 18.773 MeV 7 Be 7 Be + e− e → 7 Li + νe Q = 0.862 MeV (89.7%) Q = 0.384 MeV (10.3%) 8B 8 B → 8 Be∗ + e+ + ν e Q ≤ 15 MeV CNO 13 N → 13 C + e+ + ν e Q ≤ 1.199 MeV CNO 15 O → 15 N + e+ + ν e Q ≤ 1.732 MeV CNO 17 F → 17 O + e+ + ν e Q ≤ 1.740 MeV • Six of the reactions produce spectra with a range of Q-values and two are line sources. • Neutrinos from the CNO reactions are difficult to detect because – they are weak (less than 2% of the Sun’s energy comes from the CNO cycle) – the energies are low. • Therefore, our primary concern will be with the first five reactions, which correspond to steps of the PP chains. • The solar neutrino spectrum predicted by the Standard Solar Model is shown in Fig. 9.6 and Fig. 9.7 illustrates the radial regions of the Sun responsible for producing neutrinos from each of the PP reactions. 9.3. SOLAR NEUTRINO PRODUCTION Ga 1012 355 Cl Kamiokande pp Spectrum of Solar Neutrinos 13N 108 15O 106 8B 17F pep Flux at 1 AU (cm-2 s-1) 1010 102 0.1 hep 7Be 7Be 104 0.2 0.5 1 2 5 10 20 Neutrino Energy (MeV) Figure 9.6: The solar neutrino spectrum. The sensitive region of various experiments is indicated above the graph. The solar neutrino spectrum predicted by the Standard Solar Model is shown in Fig. 9.6. CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 356 20 8B dQ / d (R/R ) 15 7Be 10 Differential luminosity pp hep 5 0 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 R/R Figure 9.7: Differential neutrino production as a function of solar radius. The shaded area indicates the differential photon luminosity. Fig. 9.7 illustrates the radial regions of the Sun responsible for producing neutrinos from each of the PP reactions. 9.3. SOLAR NEUTRINO PRODUCTION 357 20 8B dQ / d (R/R ) 15 7Be 10 Differential luminosity pp hep 5 0 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 R/R • Notice from the above figure that the 8 B and 7Be neutrinos probe much smaller radii than the photons or neutrinos produced in PP-I (labeled pp), because they are produced at higher temperatures. • Attempts to understand the observed rate of neutrino emission from the Sun yielded initially surprising results suggesting that our fundamental understanding of either (or both) elementary particle physics and the way the Sun works were incomplete. • Resolution of this issue has led to profound new understanding in both astrophysics and elementary particle physics. We will discuss that in more depth in the next chapter. 358 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE 9.4 Evolutionary Timescales A question of basic importance is how long a star will remain on the main sequence. • Evolution prior to the main sequence (protostar stage) is governed by two primary timescales: 1. The hydrodynamical (free-fall) timescale 2. The Kelvin–Helmholtz (thermal adjustment) timescale. • Evolution on the main sequence and beyond is governed in addition by a third timescale, the nuclear burning timescale. • The hydrodynamical timescale is typically hours to days for most stars. • The Kelvin–Helmholtz timescale is typically hundreds of thousands to hundreds of millions of years. • The nuclear burning timescale depends on the fuel being burned and the mass of the star (among other factors), but is typically much longer than the hydrodynamic and Kelvin–Helmholtz timescales. • Thus, stars spend much more time on the main sequence than in their formation phase because time spent on the main sequence is governed by the hydrogen burning timescale, which is much longer than the hydrodynamical and Kelvin–Helmholtz timescales. 9.4. EVOLUTIONARY TIMESCALES For the Sun • the hydrodynamical timescale is about an hour, • the Kelvin–Helmholtz timescale is about 10 million years, and • the time to burn the core hydrogen fuel on the main sequence (nuclear burning timescale) is about 10 billion years. 359 360 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE Once stars exhaust their core hydrogen and leave the main sequence, they can undergo successive burnings of heavier fuels, which introduce new nuclear burning timescales. • In the periods between exhaustion of one fuel and ignition of another, thermal adjustment timescales will also be important. • In certain cases (such as gravitational core collapse) hydrodynamical timescales will be relevant. • The nuclear burning timescales that become important after the main sequence are typically longer than the corresponding Kelvin–Helmholtz and hydrodynamical timescales, just as for the main sequence. • However, post main-sequence burning timescales are much shorter than that for main-sequence hydrogen burning because they necessarily occur at much higher temperatures and densities. • Thus, a star generally spends more time on the main sequence than in its post main-sequence evolution. • We conclude that the nuclear burning timescale on the main sequence is longer than any other timescale in a star’s life. • Thus at any one time in a population of stars we expect to see the majority on the main sequence • (Unless the age is sufficiently large that most stars have had time to evolve off the main sequence). 9.5. EVOLUTION OF STARS ON THE MAIN SEQUENCE 9.5 Evolution of Stars on the Main Sequence The main sequence is the longest and most stable period of a star’s life but stars do evolve on the main sequence, primarily in response to core concentration changes as they burn hydrogen to helium in hydrostatic equilibrium. • This lowers the pressure in the core because it increases the mean molecular weight µ : P=ρ kT µ • This in turn increases the core density and releases gravitational energy, half of which is radiated away and half of which raises the core temperature (the virial theorem). • Because of the higher core temperature the energy outflow causes the outer layers to expand slightly and the star becomes more luminous in response to the core increase in temperature. • Dependence of luminosity on the mean molecular weight is strong, varying as approximately the 7.5 power of the mean molecular weight (Exercise 9.1). 361 362 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE • The surface temperature during evolution on the main sequence may either increase or decrease. – For stars below about 1.25 M⊙ the surface temperature tends to increase. – For more massive stars it tends to decrease as the star evolves on the main sequence. • Therefore, the primary external effect of a star’s evolution on the main sequence is to cause a small drift from the ZAMS position in the HR diagram: – Slightly upward and to the left for lighter stars. – Slightly upward and to the right for heavier stars. • Internally the changes are more substantial, but their effects are often not very visible externally while the star continues to burn core hydrogen. • Significant modification of elemental abundances is taking place as a result of the core fusion, but these changes are limited initially to the central regions. 9.5. EVOLUTION OF STARS ON THE MAIN SEQUENCE The Standard Solar Model indicates that over the 4.6 billion year time that the Sun has spent on the main sequence • The radius has increased by about 12%, • The core temperature has increased by about 16%, • The luminosity has increased by about 40%, • The effective surface temperature has increased by about 3%, and the flux of 8B neutrinos has increased by more than a factor of 40. • Near the center the mass fraction of hydrogen has decreased and the mass fraction of helium has increased by about a factor of 2 from their initial values, • Outside of about 20% of the solar radius hydrogen and helium retain their ZAMS abundances. • The mass fraction of hydrogen fuel has decreased substantially in the solar core over its lifetime, but the rate of energy production by the PP chain is dε ≃ ρ 2X 2T 4, dt where ρ is the density, X the hydrogen mass fraction, and T the temperature. – Increasing ρ and T more than offset decreasing X as the Sun evolves on the main sequence. – This explains why the Sun’s luminosity is rising even as its hydrogen fuel is being depleted. 363 364 CHAPTER 9. LIFE AND TIMES ON THE MAIN SEQUENCE Although the internal changes discussed in the preceding example lead to only small visible external modification of the star on the main sequence, they set the stage for rapid evolution away from the main sequence that will be the topic of subsequent chapters.